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Network associated with the set of elements and linearly ordered subset of elements, known as paths, satisfying the switching property is an abstract network. Due to the switching property, flows crossing at intersections are diverted to the non-crossing sides. Each element of an abstract network is equipped with two types of integral capacities: one is movement capacity which transships the flow from an element to its adjacent element and another is the storage capacity which holds the flow at the element. Due to insufficient movement capacity of intermediate elements, flow out from the source may not reach at the destination. If the flow out from the source is more than the minimum cut capacity, then the problem associated with the settlement of excess flow at appropriate intermediate elements is termed as network flow with intermediate storage. In this paper, we discuss the static and dynamic flow models with intermediate storage in an abstract network using temporal repetition of flow. We solve abstractmaximum dynamic flow and contraflow problems with intermediate storage.

Discrete and continuous time dynamic flow problems have been studied for decades. The purpose of the network flow problem is to find the maximum flow that can be sent from the source node to the destination node. Our aim is to review the general class of continuous time dynamic flow problems. We discuss about static cut and generalized dynamic cut, the latter one used to prove the maximum flow minimum cut theorem in continuous case.

An important aspect of evacuation planning is to save human population by sending the people in dangerous areas (sources) to the safe places (sinks). Optimization models for evacuation planning focus on regulating the tra_c ow in urban road networks so as to maximize the number of evacuees reaching the safe places or to minimize the evacuation time of evacuees. Recent studies show that reversing the direction of the usual tra_c ow in necessary road segments increases the tra_c ow and decreases the evacuation time signi_cantly. However, this may block the ow of some necessary support towards the source. Based on network ow models, we consider the problem of saving a path from a given node to the source, in a single-source-single-sink network, as a bilevel program. With an objective to minimize a function depending on ow rate and the path chosen, the upper level selects a path from a given node to the source, and the lower level maximizes the dynamic ow allowing arc reversals in the resulting network within a given time horizon. We discuss a solution strategy based on replacing the lower level problem by the corresponding KKT conditions. GANIT J. Bangladesh Math. Soc. 42.1 (2022) 010- 024

One of the challenges in operations research is to route numerous commodities from particular supply locations to the corresponding demand points across the lanes of a network infrastructure while maintaining capacity restrictions. The quickest multi-commodity flow problem would be one of those that reduces the time it takes to complete the process. Reorienting lanes toward demand sites can increase outbound lane capacity. The quickest multi-commodity contraflow problem is NP-hard computationally. We use a ∆-condensed time-expanded graph to propose an FPTAS for this problem by including the lane reversal technique. We look into asymmetric transit times on anti-parallel arcs to address the unequal road conditions and flow dependency.

In dynamic network flows, when some commodities move from one place to another, time taken by them to traverse the path are taken into account, unlike the static flows. Time being continuous entity, network flow problems are modeled in its continuous setting to get more realistic solution to the optimization problems. Rather, it can also be discretized to get more computational results. In this paper, insight is given to the models formulated by the researchers in dynamic network flows along with some practical applications regarding the triple optimization problem in continuous time setting mainly focusing on cost minimization problems, since the quickest path problem and the maximum flow problem are considered as two important special cases of the minimum cost flow problem. Different solution strategies and their efficiencies carried out by them are also observed. Further possible modification in the existing formulations and algorithms for better solution of problems are suggested as research endeavor.

In many real-world situations, there are numerous network optimization problems where the network attributes depend on time. In this paper, we consider single-source single-sink discrete-time dynamic network flow problems. We review some algorithms for the quickest flow problems in two environments (to the network attributes): time-invariant and time-variant. This paper mainly focuses on the existing algorithms for a later one. In literature, most of the authors have made their objectives to determine the earliest arrival time paths along which a given amount of flow can be sent in the minimum time. Evacuation is the most recent research area of network optimization, where quickest flow models allow the estimation of the minimum time required to bring a given number of evacuees to safety.

Multi-commodity flow problem appears when several distinct commodities are shipped from supply nodes to the demand nodes through a network without violating the capacity constraints. The quickest multi-commodity flow problem deals with the minimization of time satisfying given demand. Ingeneral, the quickest multi-commodity flow problems are computationally hard. The outbound lane capacities can be increased through reverting the orientation of lanes towards the demand nodes. We present two approximation algorithms by introducing partial contraow technique in the continuous-time quick estmulti-commodity ow problem: one polynomial-time with the help of length-bounded flow and another FPTAS by using _-condensed time-expanded graph. Both algorithms reverse only necessary arc capacities to get the optimal solutions and save unused arc capacities which may be used for other purposes.

Evacuation planning problem deals with sending the maximum number of evacuees from the danger zone to the safe zone in minimum time as eciently as possible. The dynamic network flow models for various evacuation network topology have been found suitable for the solution of such a problem. Bus based evacuation planning problem (BEPP), as an important variant of the vehicle routing problem (VRP), is one of the emerging evacuation planning problems. In this work, an organized overview of this problem with a focus on their solution status is compactly presented. Arrival patterns of the evacuees including their transshipments at different pickup locations and their assignments are presented. Finally, a BEPP model and a solution for a special network are also proposed.

Routing of more than one different commodity from specific origin nodes to the corresponding destination nodes through the arcs of an underlying network respecting the capacity constraints is one of the main problems in operational research. Among them, the quickest multi-commodity flow problem concerns with minimization of time taken to complete this process. The general multi-commodity and quickest multi-commodity flow problems are computationally hard. By flipping the orientation of lanes towards the demand nodes, the outbound lane capacities are increases. We introduce lane reversals in the quickest multi-commodity flow problem and present two approximation algorithms, one polynomial-time with the help of length-bounded flow and another FPTAS by using ∆-condensed time-expanded graph. Both algorithms prevent reversing arc capacities that are not required by the optimal flows that may be of interest for other purposes.

People have unpleasant experience of driving through congested road network of urban areas, where transit time is not constant but flow-dependent. Network flow over time with flow-dependent transit times is more realistic than flow over time with fixed transit time on arcs which is an active area of research from last two decades. In traffic assignment problem, transmission time of the vehicles directly depends on the number of vehicles entering on the road. So, general transit time function is non-decreasing, convex and left-continuous. In this paper we discuss how a general transit time function can be approximated by the step function. It relaxes the inflow-dependent transit times to a bow graph. We calculate maximum bow flow by using temporally repeated flow for the given time horizon. Our aim is to show how flow value in step function converges to the flow value of general transit time function by increasing number of points in the domain of the step function.

Despite of implicit flow conservation on every arc of traditional network flow model, the generalized network flow model assumes the proportional and symmetric loss factor on each arc. This paper considers two terminal lossy networks specifying the portion of flow entering an arc at its tail node that reaches to its head node. We studied the generalized maximum continuous dynamic contraflow (GMCDCF) and generalized continuous earliest arrival contraflow (GCEACF) problems. The problems are efficiently solved with pseudo-polynomial time algorithms. Moreover, a fully polynomial time approximation scheme (FPTAS) is proposed in polynomial time.

The multicommodity flow problem deals with the transshipment of more than one commodity from respective sources to corresponding sinks without violating the capacity constraints. Due to the capacity constraints, flows out from the sources may not reach their sinks, and so, the storage of excess flows at intermediate nodes plays an important role in the maximization of flow values. In this paper, we introduce the maximum static as well as maximum dynamic multicommodity flow problems with intermediate storage. We present polynomial and pseudopolynomial time algorithms for the former and latter problems, respectively. We also present the solution procedures to these problems in contraflow network having symmetric as well as asymmetric arc transit times. We transform the solutions in continuous-time settings by using natural transformation.

The abstract flow model is the generalization of network flow model which deals with the flow paths (routes) satisfying the switching property. Contraflow model increases the flow value by reversing the required arc directions from the sources to the sinks. In this paper, we integrate the concepts of abstract flow and contraflow to introduce abstract earliest arrival transshipment contraflow model in multi-terminal abstract network. The abstract contraflow on multi-terminal dynamic network is NP-Complete. We present an efficient approximation algorithm to solve the problem. This approach satisfies the demand of sinks by sending optimal flow at every possible time point and seeks to eliminate the crossing conflicts.

Network flow over time is an important area for the researcher relating to the traffic assignment problem. Transmission times of the vehicles directly depend on the number of vehicles entering the road. Flow over time with fixed transit times can be solved by using classical (static) flow algorithms in a corresponding time expanded network which is not exactly applicable for flow over time with inflow dependent transit times. In this paper we discuss the time expanded graph for inflow-dependent transit times and non-existence of earliest arrival flow on it. Flow over time with inflow-dependent transit times are turned to inflow-preserving flow by pushing the flow from slower arc to the fast flow carrying arc. We gave an example to show that time horizon of quickest flow in bow graph GB was strictly smaller than time horizon of any inflow-preserving flow over time in GB satisfying the same demand. The relaxation in the modified bow graph turns the problem into the linear programming problem.

The optimization models and algorithms with their implementations on flow over time problems have been an emerging field of research because of largely increasing human-created and natural disasters worldwide. For an optimal use of transportation network to shift affected people and normalize the disastrous situation as quickly and efficiently as possible, contraflow configuration is one of the highly applicable operations research (OR) models. It increases the outbound road capacities by reversing the direction of arcs towards the safe destinations that not only minimize the congestion and increase the flow but also decrease the evacuation time significantly.In this paper, we sketch the state of quickest flow solutions and solve the quickest contraflow problem with constant transit times on arcs proving that the problem can be solved in strongly polynomial time $~O(nm^2~(\log~n)^2)~$, where $~n~$ and $~m~$ are number of nodes and number of arcs, respectively in the network. This contraflow solution has the same computational time bound as that of the best min-cost flow solution. Moreover, we also introduce the contraflow approach with load dependent transit times on arcs and present an efficient algorithm to solve the quickest contraflow problem approximately. Supporting the claim, our computational experiments on Kathmandu road network and on randomly generated instances perform very well matching the theoretical results. For a sufficiently large number of evacuees, about double flow can be shifted with the same evacuation time and about half time is sufficient to push the given flow value with contraflow reconfiguration.

The very challenging emergency issues because of large scale natural or man-created disasters promote the research on evacuation planning. The earliest arrival contraflow is an important model for evacuation planning that rescue as many evacuees as possible at any point in time by reversing the direction of arcs towards the safe destinations with increased outbound arc capacity. We present efficient algorithms to solve the earliest arrival contraflow problem on multiple sources and on multiple sinks networks separately. We also introduce an approximate-earliest arrival contraflow solution on multi-terminal networks.

Dynamic network flow problems have wide applications in evacuation planning. From a given subset of arcs in a directed network, choosing the suitable arcs for facility location with a given objective is very important in the optimization of flow in emergency cases. Because of the decrease in capacity of an arc by placing a facility in it, there may be a reduction in the maximum flow or increase in the quickest time. In this work, we consider a problem of identifying the optimal facility locations so that the increase in the quickest time is minimum. Introducing the quickest FlowLoc problem, we give strongly polynomial time algorithms to solve the single facility case. Realizing NP-hardness of the multi-facility case, we develop a mixed integer programming formulation of it and propose two polynomial time heuristics for its solution. Because of the growing concerns of arc reversals in evacuation planning, we introduce the quickest ContraFlowLoc problem and present exact algorithms to solve the single-facility case and heuristics to solve the multi-facility case, with polynomial time complexity. The solutions thus obtained here are practically important, particularly in evacuation planning, to systematize traffic flow with facility allocation minimizing evacuation time.

With an increasing number of large-scale natural and man-created disasters over the last decade, there is growing focus on the application of operations research techniques for humanitarian relief in the emerging field of emergency evacuation. Even though a large diversity of models have been developed, many rely on solving network-flow problems on appropriate graphs. In this survey, we give a systematic collection of network flow models used in emergency evacuation and their applications. We especially focus on results interrelating these models. Considered models include max flows and min cost flows, lexicographic flows, quickest flows, and earliest arrival flows, as well as contraflows and time-dependent problems.

In the context of growing number of natural or man-made disasters, operations research methodologies are imperative for optimal and equitable use of resources available for saving life and relief supports. On the PPRR risk management model, preparedness or planning is most important in unavoidable disasters, as most of the damages are due to lack of proper policy and effective planning strategies for optimal use of available resources. To cope with problem of saving affected and normalizing the situation after disaster is also challenging. On the basis of the recent researches, the importance of mathematical modeling in emergency planning is highlighted. In this work, basic models for facility locations, evacuation planning, and relief distribution (humanitarian logistics) are discussed with examples. Recent trends in extending models to make them closer and closer to real-life situations are recorded with their solution strategies, applications, and case studies in brief.

The evacuation planning problem can be viewed as different variants of dynamic flow maximization and time minimization problems. An optimal solution to the latter problem sends a given amount of flow from disaster zones to safe zones in minimum time. We solve this problem on an embedded integrated network of a prioritized primary and a bus-routed secondary sub-networks. For a lexicographically maximum (lex-max) dynamic flow problem, we are given a time horizon and a prioritized network, where we need a feasible dynamic flow that lexicographically maximizes the flow amount leaving each terminal respecting the priority. Here, we use the quickest transshipment partial arc reversal strategy to collect the evacuees in minimum time from the disaster zones to the pickup locations of the primary sub-network. By treating such pickup locations as sources, the available set of transit-buses is assigned in the secondary sub-network to shift the evacuees finally to the sinks on the first-come-first-serve basis. This novel approach proposed here is better suited for the simultaneous flow of evacuees with minimum waiting delay at such pickup locations in the integrated evacuation network topology. The lane reversal strategy significantly reduces the evacuation time, whereas reversing them only partially has an additional benefit that the unused road capacities can be used for supplying emergency logistics and allocating facilities as well.